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Summary

It has been 20 years since the last edition of this classic text. Kevin Wainwright, a long time user of the text (British Columbia University and Simon Fraser University), has executed the perfect revision-he has updated examples, applications and theory without changing the elegant, precise presentation style of Alpha Chiang. Readers will find the wait was worthwhile.

Table of Contents

PART ONE INTRODUCTION

1

(29)

The Nature of Mathematical Economics

2

(3)

Mathematical versus Nonmathematical Economics

2

(2)

Mathematical Economics versus Econometrics

4

(1)

Economic Models

5

(24)

Ingredients of a Mathematical Model

5

(2)

Variables, Constants, and Parameters

5

(1)

Equations and Identities

6

(1)

The Real-Number System

7

(1)

The Concept of Sets

8

(7)

Set Notation

9

(1)

Relationships between Sets

9

(2)

Operations on Sets

11

(1)

Laws of Set Operations

12

(2)

Exercise 2.3

14

(1)

Relations and Functions

15

(5)

Ordered Pairs

15

(1)

Relations and Functions

16

(3)

Exercise 2.4

19

(1)

Types of Function

20

(5)

Constant Functions

20

(1)

Polynomial Functions

20

(1)

Rational Functions

21

(2)

Nonalgebraic Functions

23

(1)

A Digression on Exponents

23

(1)

Exercise 2.5

24

(1)

Functions of Two or More Independent Variables

25

(2)

Levels of Generality

27

(2)

PART TWO STATIC (OR EQUILIBRIUM) ANALYSIS

29

(94)

Equilibrium Analysis in Economics

30

(18)

The Meaning of Equilibrium

30

(1)

Partial Market Equilibrium---A Linear Model

31

(4)

Constructing the Model

31

(2)

Solution by Elimination of Variables

33

(1)

Exercise 3.2

34

(1)

Partial Market Equilibrium---A Nonlinear Model

35

(5)

Quadratic Equation versus Quadratic Function

35

(1)

The Quadratic Formula

36

(1)

Another Graphical Solution

37

(1)

Higher-Degree Polynomial Equations

38

(2)

Exercise 3.3

40

(1)

General Market Equilibrium

40

(6)

Two-Commodity Market Model

41

(1)

Numerical Example

42

(1)

n-Commodity Case

43

(1)

Solution of a General-Equation System

44

(1)

Exercise 3.4

45

(1)

Equilibrium in National-Income Analysis

46

(2)

Exercise 3.5

47

(1)

Linear Models and Matrix Algebra

48

(34)

Matrices and Vectors

49

(2)

Matrices as Arrays

49

(1)

Vectors as Special Matrices

50

(1)

Exercise 4.1

51

(1)

Matrix Operations

51

(8)

Addition and Subtraction of Matrices

51

(1)

Scalar Multiplication

52

(1)

Multiplication of Matrices

53

(3)

The Question of Division

56

(1)

The Σ Notation

56

(2)

Exercise 4.2

58

(1)

Notes on Vector Operations

59

(8)

Multiplication of Vectors

59

(1)

Geometric Interpretation of Vector Operations

60

(2)

Linear Dependence

62

(1)

Vector Space

63

(2)

Exercise 4.3

65

(2)

Commutative, Associative, and Distributive Laws

67

(3)

Matrix Addition

67

(1)

Matrix Multiplication

68

(1)

Exercise 4.4

69

(1)

Identity Matrices and Null Matrices

70

(3)

Identity Matrices

70

(1)

Null Matrices

71

(1)

Idiosyncrasies of Matrix Algebra

72

(1)

Exercise 4.5

72

(1)

Transposes and Inverses

73

(5)

Properties of Transposes

74

(1)

Inverses and Their Properties

75

(2)

Inverse Matrix and Solution of Linear-Equation System

77

(1)

Exercise 4.6

78

(1)

Finite Markov Chains

78

(4)

Special Case: Absorbing Markov Chains

81

(1)

Exercise 4.7

81

(1)

Linear Models and Matrix Algebra (Continued)

82

(41)

Conditions for Nonsingularity of a Matrix

82

(6)

Necessary versus Sufficient Conditions

82

(2)

Conditions for Nonsingularity

84

(1)

Rank of a Matrix

85

(2)

Exercise 5.1

87

(1)

Test of Nonsingularity by Use of Determinant

88

(6)

Determinants and Nonsingularity

88

(1)

Evaluating a Third-Order Determinant

89

(2)

Evaluating an nth-Order Determinant by Laplace Expansion

91

(2)

Exercise 5.2

93

(1)

Basic Properties of Determinants

94

(5)

Determinantal Criterion for Nonsingularity

96

(1)

Rank of a Matrix Redefined

97

(1)

Exercise 5.3

98

(1)

Finding the Inverse Matrix

99

(4)

Expansion of a Determinant by Alien Cofactors

99

(1)

Matrix Inversion

100

(2)

Exercise 5.4

102

(1)

Cramer's Rule

103

(4)

Derivation of the Rule

103

(2)

Note on Homogeneous-Equation Systems

105

(1)

Solution Outcomes for a Linear-Equation System

106

(1)

Exercise 5.5

107

(1)

Application to Market and National-Income Models

107

(5)

Market Model

107

(1)

National-Income Model

108

(1)

IS-LM Model: Closed Economy

109

(2)

Matrix Algebra versus Elimination of Variables

111

(1)

Exercise 5.6

111

(1)

Leontief Input-Output Models

112

(8)

Structure of an Input-Output Model

112

(1)

The Open Model

113

(2)

A Numerical Example

115

(1)

The Existence of Nonnegative Solutions

116

(2)

Economic Meaning of the Hawkins-Simon Condition

118

(1)

The Closed Model

119

(1)

Exercise 5.7

120

(1)

Limitations of Static Analysis

120

(3)

PART THREE COMPARATIVE-STATIC ANALYSIS

123

(96)

Comparative Statics and the Concept of Derivative

124

(24)

The Nature of Comparative Statics

124

(1)

Rate of Change and the Derivative

125

(3)

The Difference Quotient

125

(1)

The Derivative

126

(1)

Exercise 6.2

127

(1)

The Derivative and the Slope of a Curve

128

(1)

The Concept of Limit

129

(7)

Left-Side Limit and Right-Side Limit

129

(1)

Graphical Illustrations

130

(1)

Evaluation of a Limit

131

(2)

Formal View of the Limit Concept

133

(2)

Exercise 6.4

135

(1)

Digression on Inequalities and Absolute Values

136

(3)

Rules of Inequalities

136

(1)

Absolute Values and Inequalities

137

(1)

Solution of an Inequality

138

(1)

Exercise 6.5

139

(1)

Limit Theorems

139

(2)

Theorems Involving a Single Function

139

(1)

Theorems Involving Two Functions

140

(1)

Limit of a Polynomial Function

141

(1)

Exercise 6.6

141

(1)

Continuity and Differentiability of a Function

141

(7)

Continuity of a Function

141

(1)

Polynomial and Rational Functions

142

(1)

Differentiability of a Function

143

(3)

Exercise 6.7

146

(2)

Rules of Differentiation and Their Use in Comparative Statics

148

(30)

Rules of Differentiation for a Function of One Variable

148

(4)

Constant-Function Rule

148

(1)

Power-Function Rule

149

(2)

Power-Function Rule Generalized

151

(1)

Exercise 7.1

152

(1)

Rules of Differentiation Involving Two or More Functions of the Same Variable

152

(9)

Sum-Difference Rule

152

(3)

Product Rule

155

(1)

Finding Marginal-Revenue Function from Average-Revenue Function

156

(2)

Quotient Rule

158

(1)

Relationship Between Marginal-Cost and Average-Cost Functions

159

(1)

Exercise 7.2

160

(1)

Rules of Differentiation Involving Functions of Different Variables

161

(4)

Chain Rule

161

(2)

Inverse-Function Rule

163

(2)

Exercise 7.3

165

(1)

Partial Differentiation

165

(5)

Partial Derivatives

165

(1)

Techniques of Partial Differentiation

166

(1)

Geometric Interpretation of Partial Derivatives

167

(1)

Gradient Vector

168

(1)

Exercise 7.4

169

(1)

Applications to Comparative-Static Analysis

170

(5)

Market Model

170

(2)

National-Income Model

172

(1)

Input-Output Model

173

(2)

Exercise 7.5

175

(1)

Note on Jacobian Determinants

175

(3)

Exercise 7.6

177

(1)

Comparative-Static Analysis of General-Function Models

178

(41)

Differentials

179

(5)

Differentials and Derivatives

179

(2)

Differentials and Point Elasticity

181

(3)

Exercise 8.1

184

(1)

Total Differentials

184

(3)

Exercise 8.2

186

(1)

Rules of Differentials

187

(2)

Exercise 8.3

189

(1)

Total Derivatives

189

(5)

Finding the Total Derivative

189

(2)

A Variation on the Theme

191

(1)

Another Variation on the Theme

192

(1)

Some General Remarks

193

(1)

Exercise 8.4

193

(1)

Derivatives of Implicit Functions

194

(11)

Implicit Functions

194

(2)

Derivatives of Implicit Functions

196

(3)

Extension to the Simultaneous-Equation Case

199

(5)

Exercise 8.5

204

(1)

Comparative Statics of General-Function Models

205

(13)

Market Model

205

(2)

Simultaneous-Equation Approach

207

(2)

Use of Total Derivatives

209

(1)

National-Income Model (IS-LM)

210

(3)

Extending the Model: An Open Economy

213

(3)

Summary of the Procedure

216

(1)

Exercise 8.6

217

(1)

Limitations of Comparative Statics

218

(1)

PART FOUR OPTIMIZATION PROBLEMS

219

(224)

Optimization: A Special Variety of Equilibrium Analysis

220

(35)

Optimum Values and Extreme Values

221

(1)

Relative Maximum and Minimum: First-Derivative Test

222

(5)

Relative versus Absolute Extremum

222

(1)

First-Derivative Test

223

(3)

Exercise 9.2

226

(1)

Second and Higher Derivatives

227

(6)

Derivative of a Derivative

227

(2)

Interpretation of the Second Derivative

229

(2)

An Application

231

(1)

Attitudes toward Risk

231

(2)

Exercise 9.3

233

(1)

Second-Derivative Test

233

(9)

Necessary versus Sufficient Conditions

234

(1)

Conditions for Profit Maximization

235

(3)

Coefficients of a Cubic Total-Cost Function

238

(2)

Upward-Sloping Marginal-Revenue Curve

240

(1)

Exercise 9.4

241

(1)

Maclaurin and Taylor Series

242

(8)

Maclaurin Series of a Polynomial Function

242

(2)

Taylor Series of a Polynomial Function

244

(1)

Expansion of an Arbitrary Function

245

(3)

Lagrange Form of the Remainder

248

(2)

Exercise 9.5

250

(1)

Nth-Derivative Test for Relative Extremum of a Function of One Variable

250

(5)

Taylor Expansion and Relative Extremum

250

(1)

Some Specific Cases

251

(2)

Nth-Derivative Test

253

(1)

Exercise 9.6

254

(1)

Exponential and Logarithmic Functions

255

(36)

The Nature of Exponential Functions

256

(4)

Simple Exponential Function

256

(1)

Graphical Form

256

(1)

Generalized Exponential Function

257

(2)

A Preferred Base

259

(1)

Exercise 10.1

260

(1)

Natural Exponential Functions and the Problem of Growth

260

(7)

The Number e

260

(2)

An Economic Interpretation of e

262

(1)

Interest Compounding and the Function Aert

262

(1)

Instantaneous Rate of Growth

263

(2)

Continuous versus Discrete Growth

265

(1)

Discounting and Negative Growth

266

(1)

Exercise 10.2

267

(1)

Logarithms

267

(5)

The Meaning of Logarithm

267

(1)

Common Log and Natural Log

268

(1)

Rules of Logarithms

269

(2)

An Application

271

(1)

Exercise 10.3

272

(1)

Logarithmic Functions

272

(5)

Log Functions and Exponential Functions

272

(1)

The Graphical Form

273

(1)

Base Conversion

274

(2)

Exercise 10.4

276

(1)

Derivatives of Exponential and Logarithmic Functions

277

(5)

Log-Function Rule

277

(1)

Exponential-Function Rule

278

(1)

The Rules Generalized

278

(2)

The Case of Base b

280

(1)

Higher Derivatives

280

(1)

An Application

281

(1)

Exercise 10.5

282

(1)

Optimal Timing

282

(4)

A Problem of Wine Storage

282

(1)

Maximization Conditions

283

(2)

A Problem of Timber Cutting

285

(1)

Exercise 10.6

286

(1)

Further Applications of Exponential and Logarithmic Derivatives

286

(5)

Finding the Rate of Growth

286

(1)

Rate of Growth of a Combination of Functions

287

(1)

Finding the Point Elasticity

288

(2)

Exercise 10.7

290

(1)

The Case of More than One Choice Variable

291

(56)

The Differential Version of Optimization Conditions

291

(2)

First-Order Condition

291

(1)

Second-Order Condition

292

(1)

Differential Conditions versus Derivative Conditions

293

(1)

Extreme Values of a Function of Two Variables

293

(8)

First-Order Condition

294

(1)

Second-Order Partial Derivatives

295

(2)

Second-Order Total Differential

297

(1)

Second-Order Condition

298

(2)

Exercise 11.2

300

(1)

Quadratic Forms---An Excursion

301

(12)

Second-Order Total Differential as a Quadratic Form

301

(1)

Positive and Negative Definiteness

302

(1)

Determinantal Test for Sign Definiteness

302

(3)

Three-Variable Quadratic Forms

305

(2)

n-Variable Quadratic Forms

307

(1)

Characteristic-Root Test for Sign Definiteness

307

(5)

Exercise 11.3

312

(1)

Objective Functions with More than Two Variables

313

(5)

First-Order Condition for Extremum

313

(1)

Second-Order Condition

313

(3)

n-Variable Case

316

(1)

Exercise 11.4

317

(1)

Second-Order Conditions in Relation to Concavity and Convexity

318

(13)

Checking Concavity and Convexity

320

(4)

Differentiable Functions

324

(3)

Convex Functions versus Convex Sets

327

(3)

Exercise 11.5

330

(1)

Economic Applications

331

(11)

Problem of a Multiproduct Firm

331

(2)

Price Discrimination

333

(3)

Input Decisions of a Firm

336

(5)

Exercise 11.6

341

(1)

Comparative-Static Aspects of Optimization

342

(5)

Reduced-Form Solutions

342

(1)

General-Function Models

343

(2)

Exercise 11.7

345

(2)

Optimization with Equality Constraints

347

(55)

Effects of a Constraint

347

(2)

Finding the Stationary Values

349

(7)

Lagrange-Multiplier Method

350

(2)

Total-Differential Approach

352

(1)

An Interpretation of the Lagrange Multiplier

353

(1)

n-Variable and Multiconstraint Cases

354

(1)

Exercise 12.2

355

(1)

Second-Order Conditions

356

(8)

Second-Order Total Differential

356

(1)

Second-Order Conditions

357

(1)

The Bordered Hessian

358

(3)

n-Variable Case

361

(1)

Multiconstraint Case

362

(1)

Exercise 12.3

363

(1)

Quasiconcavity and Quasiconvexity

364

(10)

Geometric Characterization

364

(1)

Algebraic Definition

365

(3)

Differentiable Functions

368

(3)

A Further Look at the Bordered Hessian

371

(1)

Absolute versus Relative Extrema

372

(2)

Exercise 12.4

374

(1)

Utility Maximization and Consumer Demand

374

(9)

First-Order Condition

375

(1)

Second-Order Condition

376

(2)

Comparative-Static Analysis

378

(3)

Proportionate Changes in Prices and Income

381

(1)

Exercise 12.5

382

(1)

Homogeneous Functions

383

(7)

Linear Homogeneity

383

(3)

Cobb-Douglas Production Function

386

(2)

Extensions of the Results

388

(1)

Exercise 12.6

389

(1)

Least-Cost Combination of Inputs

390

(12)

First-Order Condition

390

(2)

Second-Order Condition

392

(1)

The Expansion Path

392

(2)

Homothetic Functions

394

(2)

Elasticity of Substitution

396

(1)

CES Production Function

397

(2)

Cobb-Douglas Function as a Special Case of the CES Function

399

(2)

Exercise 12.7

401

(1)

Further Topics in Optimization

402

(41)

Nonlinear Programming and Kuhn-Tucker Conditions

402

(10)

Step 1: Effect of Nonnegativity Restrictions

403

(1)

Step 2: Effect of Inequality Constraints

404

(4)

Interpretation of the Kuhn-Tucker Conditions

408

(1)

The n-Variable, m-Constraint Case

409

(2)

Exercise 13.1

411

(1)

The Constraint Qualification

412

(6)

Irregularities at Boundary Points

412

(3)

The Constraint Qualification

415

(1)

Linear Constraints

416

(2)

Exercise 13.2

418

(1)

Economic Applications

418

(6)

War-Time Rationing

418

(2)

Peak-Load Pricing

420

(3)

Exercise 13.3

423

(1)

Sufficiency Theorems in Nonlinear Programming

424

(4)

The Kuhn-Tucker Sufficiency Theorem: Concave Programming

424

(1)

The Arrow-Enthoven Sufficiency Theorem: Quasiconcave Programming

425

(1)

A Constraint-Qualification Test

426

(1)

Exercise 13.4

427

(1)

Maximum-Value Functions and the Envelope Theorem

428

(7)

The Envelope Theorem for Unconstrained Optimization

428

(1)

The Profit Function

429

(1)

Reciprocity Conditions

430

(2)

The Envelope Theorem for Constrained Optimization

432

(2)

Interpretation of the Lagrange Multiplier

434

(1)

Duality and the Envelope Theorem

435

(7)

The Primal Problem

435

(1)

The Dual Problem

436

(1)

Duality

436

(1)

Roy's Identity

437

(1)

Shephard's Lemma

438

(3)

Exercise 13.6

441

(1)

Some Concluding Remarks

442

(1)

PART FIVE DYNAMIC ANALYSIS

443

(212)

Economic Dynamics and Integral Calculus

444

(31)

Dynamics and Integration

444

(2)

Indefinite Integrals

446

(8)

The Nature of Integrals

446

(1)

Basic Rules of Integration

447

(1)

Rules of Operation

448

(3)

Rules Involving Substitution

451

(2)

Exercise 14.2

453

(1)

Definite Integrals

454

(7)

Meaning of Definite Integrals

454

(1)

A Definite Integral as an Area under a Curve

455

(3)

Some Properties of Definite Integrals

458

(2)

Another Look at the Indefinite Integral

460

(1)

Exercise 14.3

460

(1)

Improper Integrals

461

(3)

Infinite Limits of Integration

461

(2)

Infinite Integrand

463

(1)

Exercise 14.4

464

(1)

Some Economic Applications of Integrals

464

(7)

From a Marginal Function to a Total Function

464

(1)

Investment and Capital Formation

465

(3)

Present Value of a Cash Flow

468

(2)

Present Value of a Perpetual Flow

470

(1)

Exercise 14.5

470

(1)

Domar Growth Model

471

(4)

The Framework

471

(1)

Finding the Solution

472

(1)

The Razor's Edge

473

(1)

Exercise 14.6

474

(1)

Continuous Time: First-Order Differential Equations

475

(28)

First-Order Linear Differential Equations with Constant Coefficient and Constant Term

475

(4)

The Homogeneous Case

476

(1)

The Nonhomogeneous Case

476

(2)

Verification of the Solution

478

(1)

Exercise 15.1

479

(1)

Dynamics of Market Price

479

(4)

The Framework

480

(1)

The Time Path

480

(1)

The Dynamic Stability of Equilibrium

481

(1)

An Alternative Use of the Model

482

(1)

Exercise 15.2

483

(1)

Variable Coefficient and Variable Term

483

(3)

The Homogeneous Case

484

(1)

The Nonhomogeneous Case

485

(1)

Exercise 15.3

486

(1)

Exact Differential Equations

486

(6)

Exact Differential Equations

486

(1)

Method of Solution

487

(2)

Integrating Factor

489

(1)

Solution of First-Order Linear Differential Equations

490

(1)

Exercise 15.4

491

(1)

Nonlinear Differential Equations of the First Order and First Degree

492

(3)

Exact Differential Equations

492

(1)

Separable Variables

492

(1)

Equations Reducible to the Linear Form

493

(2)

Exercise 15.5

495

(1)

The Qualitative-Graphic Approach

495

(3)

The Phase Diagram

495

(1)

Types of Time Path

496

(2)

Exercise 15.6

498

(1)

Solow Growth Model

498

(5)

The Framework

498

(2)

A Qualitative-Graphic Analysis

500

(1)

A Quantitative Illustration

501

(1)

Exercise 15.7

502

(1)

Higher-Order Differential Equations

503

(41)

Second-Order Linear Differential Equations with Constant Coefficients and Constant Term

504

(7)

The Particular Integral

504

(1)

The Complementary Function

505

(5)

The Dynamic Stability of Equilibrium

510

(1)

Exercise 16.1

511

(1)

Complex Numbers and Circular Functions

511

(11)

Imaginary and Complex Numbers

511

(1)

Complex Roots

512

(1)

Circular Functions

513

(2)

Properties of the Sine and Cosine Functions

515

(2)

Euler Relations

517

(2)

Alternative Representations of Complex Numbers

519

(2)

Exercise 16.2

521

(1)

Analysis of the Complex-Root Case

522

(5)

The Complementary Function

522

(2)

An Example of Solution

524

(1)

The Time Path

525

(2)

The Dynamic Stability of Equilibrium

527

(1)

Exercise 16.3

527

(1)

A Market Model with Price Expectations

527

(5)

Price Trend and Price Expectations

527

(1)

A Simplified Model

528

(1)

The Time Path of Price

529

(3)

Exercise 16.4

532

(1)

The Interaction of Inflation and Unemployment

532

(6)

The Phillips Relation

532

(1)

The Expectations-Augmented Phillips Relation

533

(1)

The Feedback from Inflation to Unemployment

534

(1)

The Time Path of π

534

(3)

Exercise 16.5

537

(1)

Differential Equations with a Variable Term

538

(2)

Method of Undetermined Coefficients

538

(1)

A Modification

539

(1)

Exercise 16.6

540

(1)

Higher-Order Linear Differential Equations

540

(4)

Finding the Solution

540

(2)

Convergence and the Routh Theorem

542

(1)

Exercise 16.7

543

(1)

Discrete Time: First-Order Difference Equations

544

(24)

Discrete Time, Differences, and Difference Equations

544

(2)

Solving a First-Order Difference Equation

546

(5)

Iterative Method

546

(2)

General Method

548

(3)

Exercise 17.2

551

(1)

The Dynamic Stability of Equilibrium

551

(4)

The Significance of b

551

(2)

The Role of A

553

(1)

Convergence to Equilibrium

554

(1)

Exercise 17.3

554

(1)

The Cobweb Model

555

(4)

The Model

555

(1)

The Cobwebs

556

(2)

Exercise 17.4

558

(1)

A Market Model with Inventory

559

(3)

The Model

559

(1)

The Time Path

560

(1)

Graphical Summary of the Results

561

(1)

Exercise 17.5

562

(1)

Nonlinear Difference Equations---The Qualitative-Graphic Approach

562

(6)

Phase Diagram

562

(2)

Types of Time Path

564

(1)

A Market with a Price Ceiling

565

(2)

Exercise 17.6

567

(1)

Higher-Order Difference Equations

568

(24)

Second-Order Linear Difference Equations with Constant Coefficients and Constant Term